Gradient index type single lens

ABSTRACT

In a gradient index type single lens having a gradient index in a direction perpendicular to the optic axis thereof, the surface of the single lens on the light beam incidence side when the single lens is used at a reduced magnification forms a planar surface, the surface of the single lens on the light beam emergence side forms a convex surface and are satisfied the following conditions: 
     
         0.55≦(1-N.sub.0)·f/r.sub.2 ≦1.20 
    
     
         0.8≦d/f≦2.2 
    
     where r 2  is the radius of curvature of the surface on said light beam emergence side, d is the thickness of the single lens, N 0  is the on-axis refractive index of the single lens, and f is the focal length of the single lens.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to a gradient index type single lens suitable asthe collimator lens of a semiconductor laser or the pickup objectivelens or the like of an optical disc.

2. Description of the Prior Art

SELFOC lens (trade name) is well known as a lens having an indexgradient in direction perpendicular to the optic axis thereof, i.e.,so-called radial gradient index lens, and has been used as an erectone-to-one magnification imaging element in a copying apparatus or thelike.

In recent years, attempts have been made to use this gradient index typelens as the pickup objective lens of a digital audio disc or the like.The use of a plano-convex gradient index type lens is shown in the 4thtopical meeting on gradient-index optical imaging systems. However, inthe single lens shown therein, only the correction of sphericalaberration which is an on-axis aberration is considered. In contrast,where such lens is actually used as a pickup objective lens or acollimator lens, not only the on-axis aberration but also off-axisaberration must be well corrected.

SUMMARY OF THE INVENTION

In view of the above-noted points it is an object of the presentinvention to provide an gradient index type single lens in whichspherical aberration and sine condition are well corrected.

The single lens according to the present invention is plano-convex inshape and, when this single lens is used at a reduced magnification, thesurface thereof on the light beam incidence side forms a planar surfaceand the surface thereof on the light beam emergence side forms a convexsurface relative to the image side (the light beam emergence side), andthe above object is achieved by the single lens satisfying the followingconditions:

    0.55≦(1-N.sub.0)f/r.sub.2 ≦1.20

    0.8≦d/f≦2'2

where r₂ is the radius of curvature of said convex surface, d is thethickness of the single lens, N₀ is the on-axis refractive index of thesingle lens, said f is the focal length of the single lens. Accordingly,where the single lens according to the present invention is used as alight pickup objective lens, the convex surface thereof faces arecording medium, and where and where the single lens is used as thecollimator lens of a semiconductor laser, the convex surface thereoffaces the semiconductor laser.

Further, the single lens according to present invention satisfies thecondition that

    0.45≦|r.sub.2 /d|≦0.58,

thereby enabling better correction of aberrations.

In the single lens according to the present invention, installing thelens in the manner as described above when it is used at a reducedimaging magnification means that a parallel light beam or a light beamapproximate to a parallel light beam enters or emerges from the planarsurface of the lens.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the shape of a single lens according to the presentinvention.

FIGS. 2A and 2B show the aberrations of an embodiment of the single lensaccording to the present invention.

FIG. 3 shows the gradient index of an embodiment of the single lensaccording to the present invention.

FIG. 4 is a schematic view showing the single lens according to thepresent invention when used as the pickup lens of an optical disc.

FIGS. 5A and 5B show the aberrations of an embodiment of the single lensshown in FIG. 4.

FIG. 6 shows an example of the lens support when the single lensaccording to the present invention is used as the pickup lens of anoptical disc.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

To correct spherical aberration and sine condition, it is necessary tomake the values of tertiary of third-order spherical aberrationcoefficient and coma coefficient small.

In a radial gradient single lens wherein the refractive index N isexpressed as follows relative to the distance r from the optic axis:

    N(r)=N.sub.0 +N.sub.1r 2+N.sub.2r 4+N.sub.3r 6+N.sub.4r 8+ (1)

(N₀, N₁, N₂, N₃, N₄ . . . constant), parameters which contribute to thevalue of the tertiary aberration coefficient are N₀, N₁, N₂ and

r₁ : radius of curvature of the first surface

r₂ : radius of curvature of the second surface

d: thickness.

Of these, the on-axis refractive index N₀ can assume only a value of theorder of 1.4-1.8 and therefore, if N₀ is regarded as N₀ ≈1.6, parameterswhich contribute to the tertiary aberration coefficient are consideredto be r₁, r₂, d, N₁ and N₂.

On the other hand, the required conditions are the following three:##EQU1## and therefore, even if r₁ is limited to r₁ = from the fact thatone surface is a plane, it is anticipated that there are many solutionsof r₂, d, N₁ and N₂ which satisfy condition (2). From among these manysolutions, ones capable of correcting high-order aberrations or ones inwhich the working distance is appropriate can be selected in accordancewith the conditions of use.

Among r₂, d, N₁ and N₂, those which contribute to the paraxial amountare r₂, d and N₁ and, as shown in P. J. Sands: Jour. Opt. Soc. Am., 60,pages 1436-1442 (1970), N₂ is in a linear relation with the tertiaryaberration coefficients. Accordingly, d, N₁, and N₂ which satisfycondition (2) for a certain r₂ can be found by the following procedures:

(1) d is given arbitrarily;

(2) N₁ is found so that f=constant;

(3) N₂ is found so that I=0; and

(4) The procedures (1)-(3) are repeated with so that d varied II=0.

After the initial values of the parameters r₂, d, N₁ and N₂ have beendetermined by such procedures, each parameter may be varied as in thecase of the conventional lens design to thereby balance each aberration.

Also, by introducing the high-order coefficients N₃, N₄, . . . of thegradient index, it is possible to correct spherical aberration betterand achieve a great aperture.

The following facts have become apparent from the above-describeddesigning process.

First, for the correction of spherical aberration and sine condition, itis desirable that r₂ and d satisfy the following conditions:

    0.55≦(1-N.sub.0)·f/r.sub.2 ≦1.20    (3-1)

    0.8≦d/f≦2.2                                  (3-2)

(1-N₀)f/r₂ indicates the value of the refractive power borne by thesecond surface relative to the refractive power of the entire system,and if this value exceeds the upper limit of condition (3-1), correctionof spherical aberration will become difficult, and if this value exceedsthe lower limit of condition (3-1), the effect of coma correction by thesecond surface will not be obtained.

Also, if d exceeds the lower limit of condition (3-2), the absolutevalue of N₁ will increase in order to keep the focal length constant andthus, manufacture will become difficult and spherical aberration will beaggravated. If d exceeds the upper limit of condition (3-2), the workingdistance will decrease.

For better correction of spherical aberration and sine condition, it isdesirable that the following condition be further satisfied:

    0.4≦|r.sub.2 /d|≦0.58      (3--3)

That is, when |r₂ | increases and the refractive power by the secondsurface decreases, the refractive power which the gradient index hasmust be increased in order to keep the focal length constant, but byincreasing d with the relation of condition (3--3) and suppressing theincrease in gradient index, spherical aberration and sine condition canbe corrected well.

Embodiments of the present invention will hereinafter be described.Table 1 below shows the lens data of first to sixth embodiments of thesingle lens according to the present invention, and as shown in FIG. 1,r₁ is the radius of curvature of the planar surface, r₂ is the radius ofcurvature of the convex surface, and d is the thickness of the lens. Theradius of curvature r₁ of the planar surface is infinite N₀, N₁, N₂, N₃and N₄, as shown in equation (1), are constants which determine thegradient index of the single lens. Also, the lens data shown are thevalues when the focal length is normalized to 1. In the presentinvention, the light beam incidence side of the single lens in a casewhere a light beam travels from the planar surface toward the convexsurface of the single lens when used at a reduced magnification as shownin FIG. 1 is defined as the object side, and the light beam emergenceside of the single lens is defined as the image side and accordingly,the value of the radii of curvature of the surfaces are positive in acase where the center of curvature lies more adjacent to the image sidethan to the surfaces, and are negative in the converse case.

                  TABLE 1                                                         ______________________________________                                        Em-                                                                           bodi-                                                                         ment r.sub.2  d      N.sub.0                                                                            N.sub.1                                                                              N.sub.2                                                                             N.sub.3                                                                             N.sub.4                          ______________________________________                                        1    -0.5556  1.133  1.6   0.0561                                                                              0.9960                                                                              0.    0.                               2    -0.6479  1.286  1.6  -0.0593                                                                              0.5197                                                                              1.9460                                                                              0.                               3    -0.7789  1.577  1.6  -0.0932                                                                              0.2555                                                                              0.4958                                                                              0.                               4    -0.9578  1.895  1.6  -0.1767                                                                              0.1318                                                                              0.0597                                                                              0.3408                           5    -0.6241  1.307  1.45 -0.1587                                                                              0.3440                                                                              1.1134                                                                              0.                               6    -0.8380  1.676  1.75 -0.0574                                                                              0.2897                                                                              0.5570                                                                              0.                               ______________________________________                                    

Table 2 below shows the values of the back focus S'_(k), tertiaryspherical aberration coefficient I, coma coefficient II, astigmatismcoefficient III, Petzval sum P, distortion coefficient V, |r₂ /d| and(1-N₀)/r₂ of the respective embodiments (No. 1-No. 6) shown in Table 1when the object is at infinity.

                                      TABLE 2                                     __________________________________________________________________________    Embodi-                                                                       ment   S'.sub.κ                                                                   I   II  III P   V   |r.sub.2 /d|                                                    (1-N.sub.0)/r.sub.2                         __________________________________________________________________________    1      1.05                                                                             0.001                                                                             0.116                                                                             -0.469                                                                            0.625                                                                             0.875                                                                             0.49                                                                              1.080                                       2      0.92                                                                             0.267                                                                             -0.164                                                                            -0.187                                                                            0.626                                                                             0.619                                                                             0.50                                                                              0.926                                       3      0.91                                                                             0.063                                                                             -0.071                                                                            - 0.198                                                                           0.633                                                                             0.586                                                                             0.49                                                                              0.770                                       4      0.64                                                                             -0.022                                                                            -0.053                                                                            - 0.095                                                                           0.667                                                                             0.353                                                                             0.51                                                                              0.626                                       5      0.84                                                                             0.162                                                                             -0.163                                                                            -0.118                                                                            0.709                                                                             0.380                                                                             0.48                                                                              0.721                                       6      0.91                                                                             0.083                                                                             -0.045                                                                            -0.259                                                                            0.574                                                                             0.757                                                                             0.50                                                                              0.895                                       __________________________________________________________________________

FIGS. 2A and 2B show the aberrations of the third embodiment (No. 3). InFIG. 2A, solid line indicates spherical aberration and broken lineindicates sine condition, and in FIG. 2B, solid line indicates thecurvature of sagittal image plane and broken line indicates thecurvature of tangential image plane. FIG. 3 shows the gradient indexN(r) of the lens shown in the third embodiment in the directionorthogonal to the optic axis thereof. In FIG. 3, the ordinate representsthe refractive index N and the abscissa represents the distance from theoptic axis (r=0).

The fourth embodiment (No. 4) has a great aperture in which particularlyNA is of the order of 0.5, and is usable as the pickup objective lens orthe like of an optical disc.

As shown in FIGS. 2A and 2B, the various aberrations are well corrected,and the aberrations of the other lenses are such that NA is 0.2-0.3 andthe half field angle is of the order of 3°, which means a goodperformance.

In any of these first to sixth embodiments, as seen from Table 2,tertiary spherical aberration coefficient and coma coefficient are wellcorrected and in making the aperture of the lens great, high-orderspherical aberrations may be corrected by the control of the coefficientof high-order gradient index.

Also, in the shown embodiments, correction of spherical aberration isaccomplished by the coefficients N₂, N₃, . . . of gradient index but asimilar effect may also be obtained by introducing a non-sphericalsurface into the second surface.

This is because for the tertiary spherical aberration coefficientcreated by the index gradient, N₂ contributes in the form of N₂ ×ƒh³(x)dx and for the tertiary coma coefficient, N₂ contributes in the formof N₂ ×ƒh² (x)h(x)dx, where h(x) is the height of the paraxial on-axislight ray at a point in a heterogeneous medium and h(x) is the height ofthe paraxial principal light ray, and integration is effected in thedirection of the optic axis of the heterogeneous medium. Accordingly,these integrated values are determined by only r₁, r₂, d, N₀, N₁, theobject and the position of the entrance pupil, but if the entrance pupillies near the lens and the lens is not so long, h(x) will become a valueconsiderably smaller than h(x) and N₂ will hardly affect the comacoefficient. That is, the value of the coma coefficient is determined byonly r₁, r₂, d, N₀, N₁ and the object distance.

It is easy to obtain the correction effect of spherical aberration by N₂by the fourth-order non-spherical coefficient of the second surface, butagain in this case, the fourth-order non-spherical coefficient does notcontribute to the coma coefficient. In the stage in which sphericalaberration has been corrected, the coma coefficient has nothing to dowith the position of the enterance pupil and therefore, if the entrancepupil lies on the second surface, the contribution of the fourth-ordernon-spherical coefficient to the coma coefficient will be 0.

Such a circumstance also basically holds true with respect to high-orderaberration and therefore, the coefficients N₂, N₃, . . . of the gradientindex are nearly equivalent to the fourth-order, the sixth-order . . .non-spherical coefficients in the correction of abberrations.

FIG. 4 is a partial schematic view showing a case where the single lensof the present invention is applied as the pickup objective lens of anoptical disc. In FIG. 4, reference numeral 1 designates the single lensof the present invention, and reference numeral 2 denotes the glassplate of an optical disc. t represents the thickness of the glass plate,N_(G) represents the refractive index of the glass plate, and WDrepresents the air space between the single lens and the glass plate.

A parallel flat plate glass so disposed rearwardly of the optical systemhas the function of correcting spherical aberration to the positive, asis well known, and particularly the tertiary spherical aberrationcoefficient in increased by ##EQU2## by the parallel flat plate glass,(f is the focal length of the entire system.) Accordingly, during thedesigning of the aforedescribed single lens, it is necessary to make thespherical aberration of the single lens under-corrected by this amount,and it is desirable that |R₂ /d| assume a value somewhat greater thanthat in the aforedescribed first to sixth embodiments.

Table 3 below shows two examples (seventh and eighth embodiments) of thelens data of the single lens 1 when t=1.2 and N_(G) =1.52 and WD-0.87.

                                      TABLE 3                                     __________________________________________________________________________    Embodiment                                                                           r.sub.2                                                                            d  N.sub.0                                                                          N.sub.1 N.sub.2 N.sub.3 N.sub.4                             __________________________________________________________________________    7      -2.8811                                                                            5.242                                                                            1.651                                                                            -2.695 × 10.sup.-2                                                              1.847 × 10.sup.-3                                                               2.900 × 10.sup.-5                                                               3.700 × 10.sup.-6             8      -3.3240                                                                            5.985                                                                            1.651                                                                            -2.191 × 10.sup.-2                                                              1.134 × 10.sup.-3                                                               7.113 × 10.sup.-5                                                               1.614 × 10.sup.-5             __________________________________________________________________________

Table 4 below shows the values of the then focal length f, the airconversion back focus S'_(k) of the single lens, the tertiary aberrationcoefficients of the entire system and |r₂ /d|(1-N₀)f/r₂.

                                      TABLE 4                                     __________________________________________________________________________    Embodiment                                                                           f  S .sub.'κ                                                                  I  II  III P  V  |r.sub.2 /d|                                                    (1-N.sub.0)f/r.sub.2                        __________________________________________________________________________    7      2.67                                                                             1.66                                                                             0.039                                                                            -0.058                                                                            -0.082                                                                            0.643                                                                            0.373                                                                            0.55                                                                              0.603                                       8      3.01                                                                             1.59                                                                             0.004                                                                            -0.050                                                                            -0.070                                                                            0.647                                                                            0.341                                                                            0.56                                                                              0.590                                       __________________________________________________________________________

In the present specification, the gradient index has been represented asshown in equation (1), but it is often the case that the gradient indexis represented by an equation like N(r)² =N₀ 2{1-(gr)² +h₄ (gr⁴ +h₆(gr)⁶ + . . . } and therefore, the values of parameters N₀, g, h₄ and h₆when the gradient index of the seventh and eighth embodiments are sorepresented will be given in Table 5 below.

                  TABLE 5                                                         ______________________________________                                        Embodiment   N.sub.0                                                                              g           h.sub.4                                                                            h.sub.6                                  ______________________________________                                        7            1.651  0.1807      2.349                                                                              9.044                                    8            1.651  0.1629      2.200                                                                              3.632                                    ______________________________________                                    

FIGS. 5A and 5B show the aberrations of the single lens shown in theseventh embodiment. The solid line and broken line in FIG. 5A are thesame as those in FIG. 2A, and the solid line and broken line in FIG. 5Bare the same as those in FIG. 2B.

The application of such single lens according to the present inventioncan be easily achieved simply by selecting an embodiment having anappropriate back focus from Table 1 and correcting the aggravation ofspherical aberration by the glass plate 2.

The aberration coefficients in Table 2 and 4 and the aberration graphsof FIGS. 2 and 5 are all the values in the state in which the object isat infinity and the entrance pupil is coincident with the forwardprincipal point position.

Also, in the present invention, it is desirable in the correction ofhigh-order spherical aberrations that as shown in FIG. 3, the singlelens have a very weak negative or positive index gradient near the opticaxis of the lens. Such a gradient index can be formed by the opticalcopolymerizing method which is to be found in Y. Koike and Y. Ohtsuka:Applied Optics, 22, pages 418-423 (1983).

Also, in the ion exchange method, it is possible by causing ions havingthe effect of increasing the refractive index by a short time of the ionexchange, for example, T1+, Cs+ or the like, to be distributed in themarginal portion of the lens.

As described above, in the present invention, one end surface is planarand yet has a good performance, and this leads not only to remarkableease of the working and inspection of the lens but also to remarkablesimplification of the structure of the lens barrel.

For example, in the case of the pickup objective lens of the opticaldisc described in connections with FIG. 4, an auto-focus mechanism andan auto-tracking mechanism are usually required in order to cope withthe surface vibration and eccentricity of the optical disc. Therefore,use is made of a method of mounting an objective lens on anelectromagnetically driven movable element called an actuator and movingthe objective lens in the direction of the optic axis and in a directionorthogonal to the optic axis.

In such as case, to enhance the responsiveness of the drive, it isrequired to reduce the weight of the objective lens itself and of thelens barrel supporting the lens.

In the present invention, the objective lens is a single lens which islight in weight, and the first surface of the objective lens is planarand therefore, the lens barrel and the mechanism for mounting the lenson the actuator are remarkably simplified.

FIG. 6 shows an example of the manner in which the pickup objective lensof the optical disc in the present invention is mounted on the actuator.

Reference numeral 1 designates the single lens in the present invention,and reference numeral 3 schematically denotes the movable portion of theactuator. The first surface which is the planar surface of the singlelens 1 may simply be adhesively connected to the end surface of themovable portion.

Thus, in the single lens of the present invention, the first surfacethereof is a planar surface, whereby the working of the lens itself isremarkably easy and the structure of the lens barrel supporting the lensis remarkably simplified and made light in weight.

Also, even in the case where the lens is used with a prism or the likebeing disposed forwardly of the lens, by using the single lens of thepresent invention with its end surface being adhesively secured to thesurface of the prism, the structure of the lens barrel can be simplifiedand in addition, surface reflection can be reduced.

In the foregoing, a case where the object point exists at infinityrelative to the planar surface has been shown as an embodiment used as areduced magnification, but even if the object point lies at a finitedistance from the planar surface, the performance of the single lenswill be good if it is used at a reduced magnification.

In the present invention, spherical aberration and sine condition arecorrected by the single lens, but such a single lens is also effectivelyusable as element of a combination lens.

As described above, according to the gradient index type single lens ofthe present invention, correction of spherical aberration and sinecondition is possible, and such single lens is usable as a collimatorlens or the pickup objective lens of an optical disc.

I claim:
 1. A gradient index type single lens having an gradient indexin a direction perpendicular to the optic axis thereof, wherein thesurface of said single lens on the light beam incidence side when saidsingle lens is used at a reduced magnification forms a planar surfaceand the surface of said single lens on the light beam emergence sideforms a convex surface, said single lens satisfying the followingconditions:

    0.55≦(1-N.sub.0)·f/r.sub.2 ≦1.20

    0.8≦d/f≦2.2

where r₂ is the radius of curvature of the surface on said light beamemergence side, d is the thickness of said single lens, N₀ is theon-axis refractive index of said single lens, and f is the focal lengthof said single lens.
 2. A gradient index type single lens according toclaim 1, wherein said d and said r₂ satisfy the relation that

    0.45≦|r.sub.2 /d|≦0.58.